Répétition soutenance

Dynamique symboliquedes systemes 2D et des arbres infinis

Soutenance de these,encadree par Marie-Pierre Beal et Mathieu Sablik

Nathalie Aubrun

LIGM, Universite Paris-Est

22 juin 2011

Nathalie Aubrun (LIGM) Dynamique symbolique 2D et des arbres 22 juin 2011 1 / 24

Outline

1 Symbolic dynamicsWhat are symbolic dynamics ?SubshiftsClasses of subshifts

2 Motivation1D subshifts vs. 2D subshiftsTwo orthogonal approaches

3 A study of 2D-shiftsThe projective subactionHochman’s resultImprovement

4 Tree-shiftsTree-shift: exampleTree automata and tree-shiftsOn AFT tree-shiftsConjugacy of tree-SFT


Symbolic dynamics

Outline






Symbolic dynamics What are symbolic dynamics ?

Discrete dynamical systems

(X ,F ) is a discrete dynamical system if:X is a topological space, called the phase spaceF is a continuous map ∶ X → X

x●





x●

F(x)●





x●

F(x)●

F 2(x)●





x●

F(x)●

F 2(x)●

F 3(x)●

F 4(x)●

F 5(x)●



Coding of the orbits

X = ⋃ni=1 Xi a partition of the phase space X

a color ai associated with each Xi



Coding of the orbits

X = ⋃ni=1 Xi a partition of the phase space X

a color ai associated with each Xi

orbit (F n(x))n∈N coded by a sequence y ∈ {a1, . . . , an}N

●

● ●

●

●

●

. . .


Symbolic dynamics Subshifts

Subshifts: topological definition

A finite alphabet, AN (or AZ, AN2or AZ2

) the configurations spacethe configurations space is endowed with the prodiscrete topology ⇒compact spacenatural action of N (or Z, N2 or Z2) by translation: the shift σ

σj(x)i = xi+j for all x ∈ AN

(Topological) Definition: subshift

A subshift is a closed and σ-invariant subset of AN (or AZ, AN2or AZ2

).



Subshifts: topological definition

A finite alphabet, AN (or AZ, AN2or AZ2

) the configurations spacethe configurations space is endowed with the prodiscrete topology ⇒compact spacenatural action of N (or Z, N2 or Z2) by translation: the shift σ

σj(x)i = xi+j for all x ∈ AN

(Topological) Definition: subshift

A subshift is a closed and σ-invariant subset of AN (or AZ, AN2or AZ2

).



Subshifts: combinatorial definition

A finite alphabetpattern p ∈ AS, where S ⊂ N is finitethe pattern p = appears in the configuration

(Combinatorial) Definition: subshift

Let F be a set of finite patterns. The subshift defined by the set of forbiddenpatterns F is the set

TF = {x ∈ AN,no pattern of F appears in x} .

Proposition

The toplogical and combinatorial definitions coincide.








Proposition









Proposition



Symbolic dynamics Classes of subshifts

Classes of subshifts: Subshifts of finite type (SFT)

Sets of configurations that avoid a finite set of forbidden patterns:alphabet { , } on None forbidden pattern

{x ∈ { , }N,∃i ∈ N ∪ {+∞}, (xj = ⇔ j ≤ i)}

Definition: subshift of finite type (SFT)

A subshift of finite type (SFT) is a subshift that can be defined by a finite set offorbidden patterns.

simplest class with respect to the combinatorial definition2D-SFT ≡ Wang tilings



Classes of subshifts: Subshifts of finite type (SFT)

Sets of configurations that avoid a finite set of forbidden patterns:alphabet { , } on None forbidden pattern

{x ∈ { , }N,∃i ∈ N ∪ {+∞}, (xj = ⇔ j ≤ i)}

Definition: subshift of finite type (SFT)

A subshift of finite type (SFT) is a subshift that can be defined by a finite set offorbidden patterns.

simplest class with respect to the combinatorial definition2D-SFT ≡ Wang tilings



Classes of subshifts: Sofic subshifts

Factor map Φ ∶ AZ2 → BZ2given by a local map φ:

x ∈ AZ2Φ(x) ∈ BZ2

Definition: sofic susbhift

A sofic subshift is the factor of an SFT.

Recodings of SFT, using local rules.

In 1D, sofic subshifts are exactly those recognized by finite automata.

In higher dimension, decide wether a subshift is sofic or not is a difficult problem.































































Classes of subshifts: Effective subshifts

Definition: effective susbhiftAn effective subshift is a subshift that can be defined by a recursively enumerableset of forbidden patterns.

reasonnable susbhiftthis class naturally appears as projective subactions of 2D-SFT



Classes of subshifts: Effective subshifts

Definition: effective susbhiftAn effective subshift is a subshift that can be defined by a recursively enumerableset of forbidden patterns.

reasonnable susbhiftthis class naturally appears as projective subactions of 2D-SFT


Motivation

Outline






Motivation 1D subshifts vs. 2D subshifts

1D subshifts vs. 2D subshifts

1D-subshifts 2D-subshiftsEmptyness of SFT ✓ ×

Periodicity in SFT∀ SFT,∃ periodic configuration ∃ aperiodic SFT

Decomposition theorem sofic subshifts

Conjugacy of SFTN ∶ ✓Z ∶ ? ×

Recognizers for SFTfinite

local automataRecognizers

for sofic subshifts finite automata textile systems

✓: decidable problem×: undecidable problem?: open problem


Motivation Two orthogonal approaches

Two orthogonal approaches

Two attempts to understand 2D-subshifts :

study 2D-SFT through operations acting on themstudy subshifts defined on a structure between dimensions 1 and 2




Two attempts to understand 2D-subshifts :study 2D-SFT through operations acting on them

study subshifts defined on a structure between dimensions 1 and 2




Two attempts to understand 2D-subshifts :study 2D-SFT through operations acting on themstudy subshifts defined on a structure between dimensions 1 and 2



Operations on 2D-SFT

factor map operation (Fact): SFT ↝ sofic subshifts

operations that preserves SFT: product (P), finite type (FT) and spatialextensionoperation mainly studied: projective subaction (SA)




factor map operation (Fact): SFT ↝ sofic subshiftsoperations that preserves SFT: product (P), finite type (FT) and spatialextension

operation mainly studied: projective subaction (SA)




factor map operation (Fact): SFT ↝ sofic subshiftsoperations that preserves SFT: product (P), finite type (FT) and spatialextensionoperation mainly studied: projective subaction (SA)



Tree-shifts

Structure in-between N and N2 ?

N N2



Tree-shifts

Structure in-between N and N2 : free semi-group with two generators M2.

N M2 N2


A study of 2D-shifts

Outline






A study of 2D-shifts The projective subaction

The projective subaction

Idea: study subdynamics of 2D-subshifts.Principle: given a subgroup G of Z2 and a 2D-susbhift T on alphabet A, andconsider the G-subshift defined by

SAG (T) = {y ∈ AG ∶ ∃x ∈ T such that y = xT} .

G = {(x , y) ∈ Z2 ∶ y = x}

Proposition (A.&Sablik)

The class of effective subshifts is stable under projective subaction.






G = {(x , y) ∈ Z2 ∶ y = x}








G = {(x , y) ∈ Z2 ∶ y = x}








G = {(x , y) ∈ Z2 ∶ y = x}








G = {(x , y) ∈ Z2 ∶ y = x}





Projective subactions of 2D-SFT

Pavlov and Schraudner (2010):what can be realized as projective subactions of 2D-SFT ?

any sofic Z-subshift of positive entropyany zero-entropy sofic Z-subshift, with some conditions on its periods

there exist classes of Z-subshifts which are not realizable as projectivesubdynamics of any Zd SFT.

⇒ no complete characterization projective subactions of 2D-SFT.
















A study of 2D-shifts Hochman’s result

Hochman’s result

. . . no complete characterization projective subactions of 2D-SFT. . .

Consider factor map operations in addition to projective subactions.

Theorem (Hochman 2010)

Any effective Zd -subshift may be obtained by SA and Fact operations on aZd+2-SFT.

Natural question: is it possible to use one dimension less ?



Hochman’s result








Hochman’s result







A study of 2D-shifts Improvement

Improvement of Hochman’s result

Theorem


Two different proofs:Durand, Romaschenko & Shen 2010, using self-similar tilingsA.& Sablik 2010, adaptation of Robinson’s construction

Many applications:correspondance between an order on subshifts and an order on languagesmultidimensional effective subshifts are sofic (A.& Sablik 2011)construction of a tiles set whose quasi-periodic tilings have a non-recursivelybounded periodicity function (Ballier& Jeandel 2010). . .




Theorem







Theorem





Tree-shifts

Outline






Tree-shifts Tree-shift: example

Example of tree-shift

alphabet A = { , }forbidden patterns: paths containing an even number of between two


Tree-shifts Tree automata and tree-shifts

Tree automata

alphabet A = { , }tree automaton A with states Q = {q0,q1,q●}transition rules:

∶ q1

q0 q0

∶ q0

q1 q1

∶ q0

q●,q1 q●,q1

∶ q●

q●,q1 q●,q1

Accepted trees:



Tree automata

alphabet A = { , }tree automaton A with states Q = {q0,q1,q●}transition rules:

∶ q1

q0 q0

∶ q0

q1 q1

∶ q0

q●,q1 q●,q1

∶ q●

q●,q1 q●,q1

Accepted trees:



Tree automata and tree-shifts

(finite) tree automata, all states are acceptinga tree is accepted iff there exists a computationdeterministic tree automata ≡ non deterministic tree automatathe set of trees accepted by a tree automaton is a subshift

Proposition (A. & Beal 2009)

A tree-shift is a sofic tree-shift iff it is recognized by a tree automaton.A tree-shift is a tree-SFT iff it is recognized by a local tree automaton.



Tree automata and tree-shifts

(finite) tree automata, all states are acceptinga tree is accepted iff there exists a computationdeterministic tree automata ≡ non deterministic tree automatathe set of trees accepted by a tree automaton is a subshift

Proposition (A. & Beal 2009)

A tree-shift is a sofic tree-shift iff it is recognized by a tree automaton.A tree-shift is a tree-SFT iff it is recognized by a local tree automaton.



Main difference with finite automata on words

Synchronizing block: every calculation (there exist at least one) of A on this blockends in the same state.

Proposition

Any finite automaton on words has a synchronizing word.

But...

Proposition (A.& Beal 2009)

There exists a deterministic, minimal and reduced tree automaton which has nosynchronizing block.



Main difference with finite automata on words

Synchronizing block: every calculation (there exist at least one) of A on this blockends in the same state.

Proposition

Any finite automaton on words has a synchronizing word.

But...


There exists a deterministic, minimal and reduced tree automaton which has nosynchronizing block.



The context automaton

The context automaton of a tree-shift T is the deterministic tree automatonC = (V ,A,∆) where

V is the set of non-empty contexts of finite blocks appearing in Ttransitions are (contT(u), contT(v)), a → contT(a,u, v), with u, v ∈ L(T).


The context automaton of a sofic tree-shift is synchronized.


The context automaton of a sofic tree-shift T has a unique minimal,irreducible and synchronized component S, called the Shannon cover of T.The Shannon cover of a sofic tree-shift is computable.




















Tree-shifts On AFT tree-shifts

On AFT tree-shifts

AFT tree-shifts: class in-between tree-SFT and sofic tree-shiftsgeneralization of AFT 1D-susbhifts, used for coding purposes

AFT tree-shifts: factors of tree-SFT, where the factor map satisfies syntacticproperties (right-resolving, left-closing and having a resolving block)theses properties on the factor map are computable


It is decidable to say wether a sofic tree-shift is AFT or not.



On AFT tree-shifts

AFT tree-shifts: class in-between tree-SFT and sofic tree-shiftsgeneralization of AFT 1D-susbhifts, used for coding purposesAFT tree-shifts: factors of tree-SFT, where the factor map satisfies syntacticproperties (right-resolving, left-closing and having a resolving block)theses properties on the factor map are computable





On AFT tree-shifts

AFT tree-shifts: class in-between tree-SFT and sofic tree-shiftsgeneralization of AFT 1D-susbhifts, used for coding purposesAFT tree-shifts: factors of tree-SFT, where the factor map satisfies syntacticproperties (right-resolving, left-closing and having a resolving block)theses properties on the factor map are computable




Tree-shifts Conjugacy of tree-SFT

The conjuguacy problem for tree-SFT

two subshifts are conjugate if they are both factor of the otherconjugate subshifts are the same, up to a recoding

Theorem (A.& Beal 2010)

The conjugacy problem is decidable for tree-SFT.

every conjuguacy can be splitted into a sequence of elementary conjuguacieselementary conjugacies ⇒ unique minimal amalgamation of a tree-SFTtwo tree-SFT are conjugate iff they have the same minimal amalgamation
















Conclusion

Conclusion

Many properties on 2D-SFT are uncomputable. . .. . . nevertheless it is possible to partially describe them.

Tree-shifts are very similar to N-subshifts. . .. . . how to characterize monoids with same properties as 1D-subshifts ?

Thank you, Спасибо, Kiitos, Merci !


Conclusion

Conclusion

Many properties on 2D-SFT are uncomputable. . .. . . nevertheless it is possible to partially describe them.Tree-shifts are very similar to N-subshifts. . .. . . how to characterize monoids with same properties as 1D-subshifts ?



Conclusion

Conclusion

Many properties on 2D-SFT are uncomputable. . .. . . nevertheless it is possible to partially describe them.Tree-shifts are very similar to N-subshifts. . .. . . how to characterize monoids with same properties as 1D-subshifts ?



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